Computing multiple solutions from knowledge of the critical set

TL;DR

This paper introduces a geometric model for analyzing functions between spaces of the same dimension, aiding in the computation of multiple solutions for nonlinear equations, with applications to differential operators and numerical methods.

Contribution

It develops a geometric framework for understanding solution sets of nonlinear functions, applicable to infinite-dimensional spaces and semilinear elliptic operators, enhancing solution computation techniques.

Findings

Model applies to a large class of semilinear elliptic operators

Demonstrates solution computation in low and high-dimensional examples

Provides insights into the global geometry of nonlinear functions

Abstract

{We explore a simple {\it geometric model} for functions between spaces of the same dimension (in infinite dimensions, we require that Jacobians be Fredholm operators of index zero). The model combines standard results in analysis and topology associated with familiar global and local aspects. Functions are supposed to be proper on bounded sets. The model is valid for a large class of semilinear elliptic differential operators. It also provides a fruitful context for numerical analysis. For a function $F: X \to Y$ between real Banach spaces, continuation methods to solve $F(x) = y$ may improve from considerations about the global geometry of $F$. We consider three classes of examples. First we handle functions from the Euclidean plane to itself, for which the reasoning behind the techniques is visualizable. The second, between spaces of dimension 15, is obtained by discretizing a nonlinear Sturm-Liouville problem for which special right hand sides admit abundant solutions. Finally, we compute the six solutions of a semilinear elliptic equation $-\Delta u - f(u) = g$ studied by Solimini.}